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Math 2214, Test 2

1. a) Find the general solution of the equation

\begin{displaymath}y''-y=0.\end{displaymath}

b) Use the method of undetermined coefficients to find a particular solution of the equation

\begin{displaymath}y''-y=e^t.\end{displaymath}

c) Find the solution of the equation in part b which satisfies the initial condition

\begin{displaymath}y(0)=2,\quad y'(0)=0.\end{displaymath}


2. Find the general solution of the equation

\begin{displaymath}y''+3y'+2y=\exp(e^t).\end{displaymath}

Hint: A substitution may help with integrals that will arise.
3. A mass of 10 g stretches a spring by 5 cm in equilibrium.
a) Find the spring constant and the equation governing the motion of the mass (use $g=980 cm/sec^2$), assuming there is no damping in the system.
b) The mass spring system above is forced with the force

\begin{displaymath}F(t)=10\cos (12t),\end{displaymath}

where the force is measured in gcm/sec$^2$. At $t=0$, the motion starts from the equilibrium position with zero velocity. Find the motion of the mass for $t>0$.
Answers:
1. a) $y=c_1e^t+c_2e^{-t}$.

b) $y_p=te^t/2$.

c) $y=(5e^{-t}+3e^t+2te^t)/4$.
2. $y=\exp(e^t-2t)+c_1e^{-2t}+c_2e^{-t}$.
3.a) $k=1960$ g/sec$^2$.

b)

\begin{displaymath}10u''+1960u=10\cos(12t),\quad u(0)=u'(0)=0.\end{displaymath}


\begin{displaymath}u(t)=(\cos(12t)-\cos(14t))/52.\end{displaymath}


next up previous
Next: About this document ... Up: 2214samp2 Previous: 2214samp2
Michael Renardy 2004-09-21